Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,22)(23,24)(26,29)(27,31);(2,4)(6,7)(9,11,14,16)(10,15,12,13)(18,22)(20,24)(27,31)(30,32);(3,8)(6,7)(10,12)(11,16)(17,19,21,23)(18,20,22,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,15,16,13)(17,19)(18,20)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,6,8,7)(9,13,14,15)(10,11,12,16)(17,20)(18,23)(19,22)(21,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=h,c^2=fghi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3238765);